Contraction criteria for incremental stability of differential systems with discontinuous right-hand sides

Incremental stability analysis, which plays a crucial role in dynamical systems, especially nonlinear systems, has attracted more and more concern for its applications in real world control systems nowadays. This paper presents a constructive approach to derive sufficient conditions for incremental exponential stability of the Filippov solutions of a class of differential systems with discontinuous right-hand sides, by introducing a sequence of continuous dynamical systems which is uniformly contracting and approximating the Filippov systems in terms of the evolution map graphs. Afterwards, several applications of the derived theoretical results are explored. Some specific classes of control dynamical systems with discontinuous right-hand sides are studied and relative detailed conditions are presented to show the power of the present approach to investigate the stability of switched dynamical systems, Hopfield neural network with discontinuous activations and sliding mode control.


Introduction
Stability analysis researches the long-term behavior of systems' steady states, which nowadays plays a more and more important role in complex systems science. In 2004, Incremental stability [1,2] has been presented as an excellent instrument for stability analysis, which is able to address problems of synchronization of coupled systems.
A dynamic system is said to be incrementally stable if its trajectories with different initial states converge to each other as time goes to infinity. Incremental stability problems attract more and more concern in recent years for the potential applications in some frontier fields, such as PI controlled missile [3] and some problems about synchronization of dynamical networks [4,5,6]. Several literatures [7,8] discussed the problems along these lines, and some examples of the applications are included in [9].
Then people encounter the problem of testing incremental stability properties. It is proved that the method based on Lyapunov function ✩ This work is jointly supported by the National Natural Science Foundation of China under Grant (No. 62072111), Shanghai Municipal Science and Technology Major Project under Grant 2018SHZDZX01 and the ZJLab, and the Science and Technology Commission of Shanghai Municipality (No. 19JC1420101). We are grateful to Christian Schulz, On Ching Lo, Fiona Ye, Harry McGee and two anonymous reviewers for helpful comments and earlier drafts. Also, we thank Professor Liu Bo for his assistance with some preliminary theories. Exponential stability, which is a special case of stability problems and has attracted a lot of interests for its wide applications in complex systems science, including some specific networks, e.g., memristorbased recurrent neural networks [16], which plays a vital part in associative memory and optimistic computation. In these specific applications, we need to guarantee exponential stability for the Filippov systems.
In recent few years, there has been growing concerns on stability analysis of Filippov systems and a number of related works and findings arising. Under the circumstance that the value function associated to the minimization problem is locally Lipschitz continuous, sufficient conditions for local stability in the sense of Filippov solutions were given in [17]. The concept of the Filippov solution was exploited to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand sides [18]. Several criteria were derived to ensure the global asymptotic stability of the error system for the delayed neural networks with discontinuous activation functions under the framework of Filippov solution [19], and detailed criteria were proposed to guarantee the exponential stability of switched Filippov systems using approximation [20].
This paper uses approximation method to research the criteria for contraction property of Filippov solutions of a class of dynamical systems with discontinuous right-hand sides. The related criteria, which is proved sufficient for exponential stable systems, is proposed here. Several primary conditions are listed and need to be satisfied in order to guarantee local existence and uniqueness of their Filippov solution. Under the conditions mentioned above, by approximating the system with a sequence of continuously differential systems, a series of conditions are proposed so that the Filippov system is exponentially stable, that is, its solutions consequently converge to one another exponentially. Through this method, sufficient conditions are proposed for exponential convergence of the Filippov systems. And then, with the theorem we proved, several applications to specific models are given. Corresponding detailed conditions for the systems' exponential stability under several specific situations are listed and several numerical examples are shown to support our conclusions afterwards.
The paper is organized as follows. In Section 2, our model formulation and the assumptions for the existence and uniqueness of the Filippov solution of the dynamical systems with discontinuous righthand sides are presented. In Section 3, we will introduce the contraction theory, on the basis of which we proposed our main theorem, researching the conditions for incremental stability of the differential systems with discontinuous right-hand sides. In Section 4, several typical applications of the main theory are presented, and relative conclusions on several kinds of systems are provided to demonstrate our main theorem.
Illustrative numerical examples are shown to belabor the effectiveness of our results in Section 5.

Preliminary
This section focuses on the model formulation of Filippov systems with respect to the dynamical systems with discontinuous right-hand sides. A special class of the systems, the switched system, is also modeled for specific analysis afterwards. Before our analysis, we need to guarantee that the Filippov solutions of the Cauchy problem of (2) have a unique solution. So several assumptions are proposed here.

Model formulation
A Filippov system can be formulated as follows.
where ∈ ℝ , the function ∶ ℝ × [0, +∞) → ℝ is discontinuous w.r.t. and . The solution of the system (1) can be defined as a solution of the following differential inclusion (Page 101-103 in [21]), with the initial state (0) = 0 :̇∈ in which where (⋅) represents the Lebesgue measure, ( , ) = { ∶ | − | ≤ } represents the -neighborhood of with the given vector norm | ⋅ |, and represents convex closure. For some ⊂ ℝ , let | | = sup ∈ | |. Then we consider the switched system as follows, which is a special class of dynamical systems with discontinuous right-hand sides, as given in Sec. 2 of [22]. For more details, the readers are referred to Chapter 2 of [15] and Page 101-103 in [21] directly.

Existence and uniqueness conditions
Before our main result, we need to first ensure existence and uniqueness of the Cauchy problem of Filippov system (2). Herein, we present several existing results on this issue. Readers are referred to [15,22,23] for the details. First, let ( , ) = [ ]( , ). 'Upper semicontinuity' for the set-valued map is defined as follows.
From Definition 4 and 5 in [22], the following assumption should be firstly satisfied so that the existence of the solution of (2) is guaranteed: is non-empty, bounded, convex and closed, and is upper semicontinuous at ( , ).
Thus, if Assumption 1 is satisfied, then system (2) has at least one solution that can be extended to ℝ + for any initial value at 0 = 0 (Chapter 2 in [15]).
Furthermore, under the conditions in Assumption 1, the following assumption guarantees that the Filippov solution of (2) is unique.
For the switched system formulated as (3), an alternative sufficient condition was presented by Theorem 2, Sec. 10, Chapt. 2 in [15] to guarantee the uniqueness of the Filippov solution.
For more general switched systems defined as (3), we need to prove the uniqueness of the solution with concrete analysis. The following lemma may help. Then under the simplest convex definition (Page 50 in [15]), the equation ̇= ( , ) has right uniqueness in the domain .
We take a simple model with two switching surfaces intersecting each other for example and research the conditions for uniqueness in Appendix A for illustration.

Contraction analysis for the Filippov system
In this section, our main theorem is expounded, together with the detailed proof. Several basic definitions for incremental exponential stability and relative primary contraction theory are introduced first, which is the foundation for the main theorem.

Incremental stability and contraction theory
Here we present some primary definitions and introduce the contraction properties and relative theory for differential systems. Before contraction analysis, firstly, we here introduce the definition of matrix measure, denoted by the function (⋅) ∶ ℝ × → ℝ. Definition 2 (Definition 1 in [20]). For any real matrix ∈ ℝ × and a given norm | ⋅ |, we define the corresponding matrix measure ( ) as The matrix measure above can be thought of as the one-sided directional derivative of the induced matrix norm function | ⋅ |, evaluated at the point , in the direction of .
Consider the ordinary differential system as follows, which is generally time-dependent: Consider the differential equation (4) and two of its solutions ( ) = ( , 0 , 0 ) and ( ) = ( , 0 , 0 ) with initial time 0 and initial value 0 and 0 respectively. We have the following definition of incremental exponential stability (IES): [20], detail in [12]). Let  ⊂ ℝ be a forward invariant set for (4) and | ⋅ | be some norm on ℝ . The system (4) is called incrementally exponentially stable (IES) in  if there exist constants > 0 and ≥ 1 such that

Remark 1.
The definition IES here is independent of the initial time 0 , i.e., the constants and are independent of 0 .

Remark 2.
There are some other types of stability property, for instance, incrementally asymptotical stability [20], asymptotical periodic stability [25], etc. The property IES here implies that the solutions of the system will converge towards each other at an exponential speed with regard to the difference of their initial values, which is stricter than incrementally asymptotical stability. Under the circumstance that the system is incrementally exponentially stable and has a periodic solution, the system's solutions will converge to the periodic trajectory at an exponential speed, called asymptotical periodic solutions (seen in Example 4 in [20] and system (22) in [18]). If the system has a solution converging to an equilibrium, the property 'incremental exponential stability' implies that the dynamics with other initial values will converge to the equilibrium as well (seen in Example 1 and 2 in Section 5 in this paper). If the system has a solution converging to the discontinuous surface, e.g. system (25), with the property 'incremental exponential stability', the dynamics with different initial values will also converge to the discontinuous surface, which helps to deal with sliding mode control problems (seen in Section 4.5 and Example 3 in Section 5).

Definition 4 (Sec. 4.4 in
We then introduce the concept of infinitesimal contraction and the contraction theory [13] for systems with continuously differentiable right-hand sides. Denote the whole state place of system (4) by Σ, Σ ⊆ ℝ .
Definition 5 (Sec. 1.1 in [13]). The continuously differentiable vector field defined by ( , ) in (4) is said to be (infinitesimally) contracting in a -reachable set  ⊂ Σ if there exists some norm in , with associated matrix measure , such that, for some constant > 0 (the contraction rate), The theory of contraction analysis states that, if a system is contracting as defined above, then all of its trajectories are incrementally exponentially stable (IES), as follows: Lemma 4 (Theorem 1 in [13]). Suppose that  is a K-reachable forwardinvariant subset of Σ and that the vector field defined by ( , ) in (4) is infinitesimally contracting with contraction rate c therein. Then, for each two of (4)'s solutions ( ) = ( , 0 , 0 ) and ( ) = ( , 0 , 0 ) with 0 , 0 ∈  we have that As a result, if a system is contracting in a forward-invariant subset, then its solutions converge to a limiting trajectory. Remark 3. From Theorem 1 and Appendix B in [13], the contraction theorem is proved with 0 = 0. Thus Definition 5 and Lemma 4 hold regardless of the value of 0 . The parameters and are also independent of 0 .

Theoretical results of contraction
By the idea in the proof of Theorem 2.2 in [27], we try to construct a sequence of functions { ( , )} satisfying the conditions as follows, denoted by  1 (Σ):
Notice that with Assumption 1 and Condition 2 in  1 (Σ) above, we can find a subsequence of { ( , )} satisfying Condition 3 in  1 (Σ). So it is unnecessary to prove Condition 3 in the proof of Theorem 1 afterwards.
We have the following main theorem.
Proof. From [15], under Assumption 1 and 2, the solutions of the Cauchy problem of (2) and (5) exist and are unique respectively for ∈ ℝ + . For any two of the solutions of (2), ( ) and ( ) with different initial values ( 0 ) = 0 and ( 0 ) = 0 , we construct solutions of (2) from sequences of solutions of (5), denoted by and respectively. With Meanwhile, since is continuous and continuously differentiable with respect to and continuous w.
Since function sequence { ( , )} is uniformly contracting, from Lemma 4, it can be shown that there exists some = > 0 such that This completes the proof. □ Remark 4. One of the essential conditions for the contraction of Filippov system (4) is the existence and uniqueness of the solution of the underlying differential inclusion. (2) is regarded as a bimodal switched system, we can replace Assumption 2 with the conditions in Assumption 3 here in Theorem 1, which is applied in the following corollaries and examples. For more general switched systems defined as (3), by satisfying the conditions in Lemma 3 or proving the solution passes across the common boundary as illustrated in Corollary 1 in Sec. 10, Chapter 2 in [15], right uniqueness of Filippov solution for the switched system (3) can be proved in the whole domain.  Herein, we present several tips for how to construct the function sequence { ( , )} for validation of the condition in Theorem 1 to guarantee that incremental stability can be obtained by the constraint ( ( , ) ) < 0. We consider two scenarios as follows.

Remark 5. If
Switched system. Consider the switched system (3) with the notations as above with necessary specifications and assume that the conditions for the existence and uniqueness of the Cauchy problem are satisfied. Let (⋅) ∶ ℝ → [0, 1] be continuous, non-decreasing, and differentiable, 1 with ( ) = 0 when ≤ − 1 2 , and ( ) = 1 when ≥ 1 2 . We also assume that each function ( , ) for each region is well-defined in the whole region. The core idea is to construct a continuous and differentiable function sequence that is a convex combination of functions ( , ) in the neighborhood region of the hypersurfaces as well as their intersections in order to converge (in graph) to the set-valued map ( )( , ).
First, we consider a simple example of switched system with a single switching hypersurface 0 = {( , ) ∶ 0 ( , ) = 0} as follows: then we construct a sequence of systems ̇= ( , ) as the convex combination of 1,2 ( , ), where A simple schematic diagram is shown in Fig. 1. It can be shown that the sequence of functions { ( , )} formulated as (7) is to be proved to approach the set-valued map as follows: 1 We highlight that (⋅) can be also continuous, non-decreasing, and piecewise differentiable. The arguments also hold. In this case, the gradient operator should be changed to Clark generalized gradient. Readers can refer to Part A in Sec. 2 in [28] and [24] for details.
For more complex circumstances, consider the switched system (3) with the notations as above and denote the whole set of switching hypersurfaces by { }. We here propose an iterative way to construct the function sequence { ( , )} similar to the processing above. Generally Let = 1∕ . So, we can get the function sequence that is continuous and differentiable in the neighborhood of * and take values as the convex combination of the values of ( , ) from the both sides of * . Hence, we iteratively consider all switching hypersurfaces in the same way and can obtain a continuous and differentiable function sequences that are convex combinations of ( , ) from both sides near each switching hypersurfaces to validate the conditions  1 (Σ). An explicit and direct way to construct | ± * can be formulated as follows. Here, In detail, we are in the stage to present the iterative method to construct the function sequence { ( , )} as follows. Let  = {1, 2, ⋯ , } be the index set of the switching hypersurfaces , = 1, ⋯ , .
Step 1: We start with a single switching hypersurface 1 and let 1 = {1} be the index set for the hypersurfaces that is considered in the iterative way. By the method mentioned in the paragraph above and Eq. (8), we are able to construct the functions with respect to the origin switching function ( , ): with | 1 ± ( , ) defined above, for example by Eq. (9).
Step 2: Iteratively, assume that for the hypersurfaces { 1 , 2 , ⋯ , −1 } with the index set −1 = {1, ⋯ , − 1}, for ≥ 1, the functions −1 are well constructed an continuous and differentiable be continuous and differentiable extension from −1 in the -neighborhood of as mentioned above. A direct and explicit version is given by (9). Then we give A simple schematic diagram for two intersecting hypersurfaces  Composite function of discontinuous functions. Here, we consider the case that discontinuity of the right-hand functions is caused by certain discontinuous function of lower-dimensions, formulated as follows. It is easy to prove that the sequence of functions ( , ) approach the right-hand side of (11) in the sense of set-valued map as converges to zero.

Remark 7.
We have presented two special scenarios on how to construct function sequence ( , ) towards satisfying the conditions of Theorem 1. However, we highlight that validation the conditions of Theorem 1 including  1 (Σ) should be further conducted.

Switched differential systems
Here we take a simple case of switched system (3) as follows: where ∈ ℝ , ∈ ℝ , with the initial default time 0 = 0, and defines the switching hyperplane with ⊤ = 1.
On the discontinuous surface ⊤ = 0, the right-hand side of the system should be consistent with the differential inclusion of the system according to (2)  Also, we need to guarantee the existence and uniqueness of the system's solution according to the conclusions in [15,23].
We have the following result:
Then we construct a sequence of maps as (7). For the system ̇= ( , ), where We have Under the condition (14) that holds for each that satisfies | ⊤ | ≤ 2 , it can be shown that ( , ) < − holds for all sufficient large . From Theorem 1, the Filippov solutions with different initial values of (12) globally exponentially converge towards each other almost everywhere in Σ. The corollary is proved. □ Corollary 2. If given that system (12) has a unique solution, then only the condition ([ 1 ( , ) − 2 ( , )] ⊤ ) ≤ 0 is necessary to prove the exponential stability in Corollary 1, that is, condition (13) is no longer needed.

Remark 8.
Notice that the switching hypersurface { ∶ ⊤ = 0} here is linear. When it comes to a system formulated like (3) with nonlinear discontinuities, a similar conclusion can be proposed and proved as well.
So, more generally, together with Remark 8, we have The detailed proof for Corollary 3 is omitted here, which is similar to the proof of Corollary 1.

Remark 10. Similar approach above has been employed in Theorem 11
in [20]. In [20], under the assumption that (12) satisfies the conditions for the existence and uniqueness of a Caratheodory solution, the conclusion is that, with + and − contracting in their domains and ([ + ( , ) − − ( , )] ⋅ ▽ ( , )) = 0 in the neighborhood of the switching hypersurface , the bimodal Filippov system is incrementally exponentially stable in a -reachable set with a specific convergence rate, which is consistent with Corollary 3.

Switched linear systems
Here we take a switched linear differential system, which is also called a discontinuous piecewise affine (PWA) system for example. More detailed conditions for exponential stability can be obtained here. Consider the PWA system as follows: with some matrices , ∈ ℝ , , vectors , , ∈ ℝ , and defines the switching hyperplane, ⊤ = 1. It is guaranteed that on the discontinuous surface ⊤ = 0, the right-hand side ( , ) ∈ { ( + ) + (1 − )( + ) ∶ ∈ [0, 1]}, which is consistent with the differential inclusion in the following proof, and the system has a unique solution according to the assumptions above.
Here we similarly construct a smooth approximation of the discontinuous right-hand side. Specific conditions for exponential stability are expounded as follows.
Similarly, we construct the following sequence of functions: with (⋅) in the form of (16), which can be shown to approach the corresponding set-valued map of (17) as → 0. The system ̇= ( ) is bounded with the bound as well. In the case | ⊤ | ≤ ∕2, we have Then under the condition (19) that , which satisfies ( ( ) ) < 0, guarantees the exponential stability, as proved in Theorem 1. □ From Corollary 4 above, we find it consistent with the following corollary, which has already been proposed in Theorem 2 in [29]: (17) is incrementally exponentially stable if there exists a positive definite matrix = ⊤ > 0, a vector ∈ ℝ and a number ∈ (0, 1) such that 1.
The detailed proof is added in Appendix B.
Remark 11. In Corollary 5, under the circumstance that = 0 which implies = , the conditions above infer that the right-hand side of system (17) is continuous. Under the circumstance that = 1, it indicates that the discontinuity may occur only due to the shift terms and .

Switched systems with hypersurfaces intersecting
In this section, we consider system (3), with the switching hypersurfaces intersecting each other. For simplicity, we here study the system as follows for example, where 1 , 2 ∈ ℝ , ∈ ℝ , and 1 , 2 define the switching hyperplanes with ⊤ 1 1 = 1 and ⊤ 2 2 = 1. And on the discontinuous surfaces, the right-hand side should be consistent with the differential inclusion of the system in the following analysis. A simple schematic diagram is shown in Fig. 3.
Proof. Similar to Corollary 1, it can be shown that the Cauchy problem of (20) has a unique solution. Then we construct a sequence of maps with the construction method above for switched systems as in Eq. (9) and (10). We first construct the following continuous functions with (⋅) defined as (16). Then, ( , ) is designed as follows, It can be shown that ( , ) is continuous and approaches the setvalued map ( , ) = [ ]( , ). The derivative of ( , ) w.r.t. , needs to satisfy that its measure ( ( , ) ) should be less than zero as → 0. which is satisfied by (21) and (22). □
For matrix = ( ), we define the matrix measure as , We can prove the following Corollary: Suppose that the solution of the Cauchy problem of (23) exists and is unique for ∈ ℝ + , and system (23) satisfies Condition  2 above, and there exists a positive diagonal matrix = { 1 , ..., }, such that holds for ∈ ℝ + . Then the Filippov solutions of (23) exponentially converge towards each other.
Proof. Under the assumption that the solution of system (23) exists and is unique for ∈ ℝ + , by the way of (11), we construct the continuous system as follows: It is easy to prove that the sequence of functions { ( )} approaches the right-hand side of (23) in the sense of set-valued map as converges to zero.
Let ̄( 0 , ) = Here we take the matrix measure ,1 (⋅), we need to ensure that As for ( ) , because (⋅) is non-decreasing, it is obvious that is a non-negative diagonal matrix.
To illustrate the application to sliding mode control, here we take a simple case for example from Section 1.3 in [32], with the positive integer ≥ 2, the -th order system is as follows.
For simplicity, let = 2, we consider the second order system: with 1 converging to zero in finite time if its trajectory reaches 1 1 + 2̇1 = 0, where 1 2 > 0 and 2 1 + 2 2 = 1. Then we seek for the qualified control law ( ) to guarantee the trajectory of (26) converges to origin in finite time.
Define coordinate with respect to stable manifold: where = ( 1 , 2 ) ⊤ , such that 1 is stable if = 0. We have the conclusion as follows.
The corollary above is easily obtained by constructing a sequence of functions satisfying Condition  1 , together with Theorem 1.
We have Thus, under the condition (29), Eq. (30) is satisfied. From Theorem 1, it is proved that the Filippov solutions of (26) exponentially converge towards each other. Considering a trajectory with its initial point satisfying (27), which obviously converges to origin in finite time, the trajectories with other initial inputs will finally converge to origin as well. The corollary is proved. □

Remark 12.
For sliding mode control problems, a stricter condition is necessary to guarantee the solutions of system (25) converge to the stable manifold in finite time. Together with the conditions above in Corollary 9, the system needs to satisfy that there exists > 0 such that Since the system is incrementally exponentially stable according to Corollary 9, the trajectory will converge to the -neighborhood of the origin in finite time, where can be arbitrarily small. Then with the new condition above, if the trajectory reaches the -neighborhood of the stable manifold ⊤ = 0, where is small enough, it is obvious that the trajectory will converge the stable manifold in finite time.
Remark 13. The spectral radius of the matrix may help deal with the conditions proposed in Corollary 9. In the continuous region, it is obvious that if the spectral radius of matrix + ℎ ( , ) , denoted by ( + ℎ ( , ) ), is less than 1, then there exists a matrix norm | ⋅ |, such can be arbitrarily small.

Numerical experiments
In this section, several numerical experiments are presented to help illustrate the main theorem and corollaries proposed above.

Example 1: one-dimensional switched systems
We take the one-dimensional switched system as an example to illustrate the main theorem. Consider the following switched system: with the sign function , which can be formulated as a differential inclusion: ̇∈ ( ) with So, it can be seen that Σ = { ∶ | | ≤ } for any ≥ 1 is invariant for this differential inclusion. Consider the following continuously differentiable function sequence: which can be shown to approach ( ) in graph as → ∞ and the differential systems: ̇= ( ) with the Jacobian: ( )∕ = −1 − (2∕ )( ∕(1 + 2 2 )), which are uniformly contracting with = −1. Hence, we can prove that ̇∈ ( ) is contracting as well.

Example 2: switched nonlinear system
Consider the following switched nonlinear system. Through the criteria proposed in Corollary 1, it can be proved that the following system is exponential stable, which is verified by numerical experiment.
Since ( ) = (0, 0, 0) ⊤ is a special solution of system (31), the solutions will finally converge to the origin as seen in Fig. 4. We find that the dynamics of system (31) converges to the discontinuous surface in finite time under the effect of the discontinuous surface since On the surface { ⊤ = 0}, according to the constraints proposed above, we have ̇( ) ∈ {(1 − 2 ) − ∶ ∈ [0, 1]}, which implies that ( ) is converging towards the coordinate origin at an exponential speed.

Conclusion and future work
In conclusions, we proposed a general criterion on the conditions for exponential incremental stability of a class of dynamical systems with discontinuous right-hand. We formulated the corresponding Filippov system for the discontinuous system and constructed a sequence of contracting dynamical systems with continuous right-hand sides to approximate the Filippov system. With the uniform contraction of the  sequence of continuous dynamical systems, the incremental stability of the Filippov system is proved in Theorem 1. To demonstrate the power of the present theory, we took several types of discontinuous dynamical systems as examples and found the criteria is efficient by selecting different matrix measures, compared with the previous ones. Meanwhile, the application to sliding mode control (SMC) problems is attached and a new train of thought is proposed, which may help construct the control law. Furthermore, how it performs compared with classical Lyapunov method for more complicate control problem related to Filippov system needs further investigation.

Author contribution statement
Lu Wenlian: conceived and designed the experiments; performed the experiments; contributed reagents, materials, analysis tools or data.
Lang Yingying: performed the experiments; analyzed and interpreted the data; contributed reagents, materials, analysis tools or data; wrote the paper.

Data availability statement
No data was used for the research described in the article.

Declaration of interests statement
The authors declare no competing interests.

Additional information
No additional information is available for this paper.

Appendix A. An example: try to prove uniqueness under given conditions with Lemma 3
Here we take a model with two switching surfaces intersecting each other and research the conditions for uniqueness for illustration.
As seem in Fig. 7, = ( , ), ∈ ⊂ ℝ 2 , = 1, 2, 3, 4, We claim that, under the condition that Assumption 3 is satisfied for points ( , ) ∈ 1 ∪ 2 , and following conditions, denoted by Condition  (33) are satisfied at the intersection of the switching surfaces = 1 ∩ For two regions and , , ∈ {1, 2, 3, 4}, if and are on the same side of the switching surface 1 (or 2 ), the uniqueness of the system's solution occurs on ∪ ∪ ( ∩ ), with Assumption 3 satisfied w.r.t. the switching surface 2 (or 1 ), according to Lemma 2,where means the boundary of region . Problem comes when we consider the intersection of the two switching surfaces = 1 ∩ 2 , which is also the common boundary of regions 1 and 4 , regions 2 and 3 .
We here take regions 1 and 4 for instance, and they have the common boundary . According to Condition  (33) given above, we have the following conditions at the point , 1.  It can be proved that if  > 0) at point , the solutions pass from 1 ∪ 2 into 3 ∪ 4 (or from 3 ∪ 4 into 1 ∪ 2 ) in the neighborhood of by virtue of Corollary 1 of Lemma 2 in Sec. 10, Chapter 2 in [15], thus we just need to analyze the conditions for uniqueness according to the switching surface 2 with Lemma 2, which is consistent to the conditions above. It is the same if  Then we need to consider the circumstance that For ∈ 1 and ∈ 4 in the neighborhood of ( , ) ∈ , from the boundedness of ∕ it follows that Similar to the proof of Theorem 2 in Sec. 10, Chapter 2 in [15], it follows from the inequalities That is, It is the same with 2 and 3 . Thus from Lemma 3, the uniqueness occurs in the neighborhood of ( , ) in domain. The proof is completed. From (36) we need 1 ( ⋅ ⊤ ) ≤ holds for | ⊤ | < 2 , which infers that Since can take any value that satisfies ⟂ , it implies that ℎ = ( − ) = ⃗ 0 together with | − 1 2 | 2 | 1 2 ( − )| 2 + ⊤ ( − ) = 0, that is, there exists > 0, such that 2 ) < 0 holds for each ∈ ℝ , which is equivalent to ( − 1 2 ) ⊤ ( + ⊤ )( − 1 2 ) < 0, that is, we need + ⊤ < 0. And it is the same with , + ⊤ < 0. With all the conditions above, we need to verify the existence and uniqueness of the system's solution. We have which is smaller than zero, and guarantees that (17) has a unique solution. □